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Properties

Label 2-740-740.607-c0-0-0
Degree $2$
Conductor $740$
Sign $0.828 - 0.559i$
Analytic cond. $0.369308$
Root an. cond. $0.607707$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.342 + 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.866 − 0.5i)5-s + (0.866 − 0.500i)8-s + (0.642 + 0.766i)9-s + (0.766 − 0.642i)10-s + (1.11 − 1.32i)13-s + (0.173 + 0.984i)16-s + (0.266 − 0.223i)17-s + (−0.939 + 0.342i)18-s + (0.342 + 0.939i)20-s + (0.499 + 0.866i)25-s + (0.866 + 1.5i)26-s + (1.92 + 0.515i)29-s + (−0.984 − 0.173i)32-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.866 − 0.5i)5-s + (0.866 − 0.500i)8-s + (0.642 + 0.766i)9-s + (0.766 − 0.642i)10-s + (1.11 − 1.32i)13-s + (0.173 + 0.984i)16-s + (0.266 − 0.223i)17-s + (−0.939 + 0.342i)18-s + (0.342 + 0.939i)20-s + (0.499 + 0.866i)25-s + (0.866 + 1.5i)26-s + (1.92 + 0.515i)29-s + (−0.984 − 0.173i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.828 - 0.559i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.828 - 0.559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7203211714\)
\(L(\frac12)\) \(\approx\) \(0.7203211714\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-0.642 + 0.766i)T \)
good3 \( 1 + (-0.642 - 0.766i)T^{2} \)
7 \( 1 + (-0.342 + 0.939i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
19 \( 1 + (-0.642 - 0.766i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.92 - 0.515i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (1.26 - 1.50i)T + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (1.58 + 1.10i)T + (0.342 + 0.939i)T^{2} \)
59 \( 1 + (0.342 + 0.939i)T^{2} \)
61 \( 1 + (0.0151 + 0.173i)T + (-0.984 + 0.173i)T^{2} \)
67 \( 1 + (0.342 - 0.939i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
79 \( 1 + (-0.342 + 0.939i)T^{2} \)
83 \( 1 + (0.984 + 0.173i)T^{2} \)
89 \( 1 + (0.939 + 0.657i)T + (0.342 + 0.939i)T^{2} \)
97 \( 1 + (0.642 - 1.11i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.49414971089005833789110612008, −9.757793998585407513330297719316, −8.471458937887760860290786296822, −8.203019984154864222752435645873, −7.36606951928213380225200249859, −6.38033566518280134667976126018, −5.24592476419512941114074158351, −4.57775991215802327480118937724, −3.38828621985639590162423941445, −1.15515603179824714247334384741, 1.34764532820558456527781263646, 2.94798617079922184621483765315, 3.93280953288788970055514309741, 4.50409029199172278294830322885, 6.30794318924991594691149348126, 7.10125698810316077412374723958, 8.174762415936907419764836702738, 8.850327134210149407259108444685, 9.774967944296272134650043742774, 10.57636020030676356897469660461

Graph of the $Z$-function along the critical line